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I believe that the diagram depicting a surjective (onto) mapping on the "Injections, Surjections, and Bijections" page contains a subtle mistake.

At first glance, this appears to fit the given definiton of surjection. But this is inconsistent with the definition of a function, given on the "Different Types of Functions" page. In particular, the aforementioned page states that a function maps every element of the domain to exactly one element in the codomain. Futhermore, the Wikipedia page on "Definition of a function" states that "Given a function f:X —> Y, the set X is the domain of f". Finally, the textbook "How to Prove It" by D. Velleman considers a relation f to be a function from A to B if for all elements in A, there exists a unique element in B such that f(a) = b.

What all of these definitions of "function" have in common is that every element of the domain gets mapped. Therefore, the mapping from A to B as depicted in the diagram above (and thus on the "Injections, Surjections, and Bijections" page) should NOT depict an element in A that isn't mapped to some element in B.

I would include an image and links in this post, but I am not allowed to because I am a "low-karma" user, presumably because I'm posting for the first time.

Administrator's Note: The image has been corrected! Once again, thank you very much for spotting this mistake!

Yes, thank you for finding this! I think the reason I made this mistake is because I was probably changing the question up and missed deleting remnants of the previous question. I also noticed that particular question can be solved much more simpler than I originally presented it to be. As a result, I tried to simplify the explanation.

Isn't it possible to construct a nonempty subset of Z that doesn't contain a least element by constructing the set of negative integers? eg, if A = { -1, -2, -3, … } then A has a largest element but does not have a least element.

Elsewhere I have seen that the well-ordering principle holds for positive integers or for nonnegative integers, but I am not sure if it holds for all integers.

Example 1
Find the most general antiderivative of f(x)=3x^^2^^.
We should be familiar with the derivative of g(x)=x^^3^^, that is g′(x)=3x^^2^^. Therefore, the most general antiderivative F(x)=3x^^2^^+C where C is any constant.

I am going to feel so dumb when this turns out to be true, but it seems to me that the most general antiderivative of f(x)=3x2 is F(x)=(3x3/3) + C or just x3 + C where C is any constant.

I would like to point out a mistake in the example on page "Increasing and Decreasing Functions"

You proved that f(x) = x^2 -ln(x) is an increasing function on (0,inf) but it's actually not.
the derivative f'(x) = 2x - 1/x and f''(x) = 2 + 1/x^2 > 0 shows that f(x) has a minimum at x0= 1/sqrt(2), f(x) is indeed decreasing on (0,x0) and increasing on (x0,inf).

Mistakes are unavoidable, but they can do nothing to make your page any less amazing, keep up the good work :). Also, it would be greatly appreciated if you can add a topic on Integral Transforms especially Laplace and Fourier transforms.